1ch FundamentalOptics Final a.qxd 6/15/2009 2:28 …... f is =for convex (diverging) ... A simple...

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Fundamental Optics

Fundamental Optics1.2

Introduction

The process of solving virtually any optical engineering problem can bebroken down into two main steps. First, paraxial calculations (first order)are made to determine critical parameters such as magnification, focallength(s), clear aperture (diameter), and object and image position. Theseparaxial calculations are covered in the next section of this chapter.

Second, actual components are chosen based on these paraxial values, andtheir actual performance is evaluated with special attention paid to theeffects of aberrations. A truly rigorous performance analysis for all but thesimplest optical systems generally requires computer ray tracing, butsimple generalizations can be used, especially when the lens selectionprocess is confined to a limited range of component shapes.

In practice, the second step may reveal conflicts with design constraints,such as component size, cost, or product availability. System parametersmay therefore require modification.

Because some of the terms used in this chapter may not be familiar to allreaders, a glossary of terms is provided in Definition of Terms.

Finally, it should be noted that the discussion in this chapter relates only tosystems with uniform illumination; optical systems for Gaussian beams arecovered in Gaussian Beam Optics.

THE OPTICALENGINEERING PROCESS

Determine basic systemparameters, such asmagnification and

object/image distances

Using paraxial formulas and known parameters,

solve for remaining values

Pick lens components based on paraxially

derived values

Estimate performance characteristics of system

Determine if chosen component values conflict

with any basic system constraints

Determine if performancecharacteristics meet original design goals

Engineering Support

CVI Melles Griot maintains a staff of knowledgeable, experiencedapplications engineers at each of our facilities worldwide. Theinformation given in this chapter is sufficient to enable the userto select the most appropriate catalog lenses for the mostcommonly encountered applications. However, when additionaloptical engineering support is required, our applications engi-neers are available to provide assistance. Do not hesitate tocontact us for help in product selection or to obtain moredetailed specifications on CVI Melles Griot products.

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1.3Fundamental Optics

Fundamental Optics

s

f

FCA

f

front focal point rear focal point

principal points

fobject Hv

image

H″F″

s″

h

h″

f = lens diameter

CA = clear aperture (typically 90% of f)

m = s″/s = h″ /h = magnification orconjugate ratio, said to be infinite ifeither s″or s is infinite

v = arcsin (CA/2s)

h = object height

h″ = image height

s = object distance, positive for object (whether real or virtual) to the left of principal point H

s″ = image distance (s and s ″ are collectively calledconjugate distances, with object and image inconjugate planes), positive for image (whether realor virtual) to the right of principal point H″

f = effective focal length (EFL) which may be positive (as shown) or negative. f represents both FH andH″F″, assuming lens is surrounded by medium of index 1.0

Note location of object and image relative to front and rear focal points.

Figure 1.1 Sign conventions

Sign Conventions

The validity of the paraxial lens formulas is dependent on adherence to the following sign conventions:

When using the thin-lens approximation, simply refer to the left and right of the lens.

For lenses: (refer to figure 1.1) For mirrors:s is = for object to left of H (the first principal point) f is = for convex (diverging) mirrors

s is 4 for object to right of H f is 4 for concave (converging) mirrors

s″ is = for image to right of H″ (the second principal point) s is = for object to left of H

s″ is 4 for image to left of H″ s is 4 for object to right of H

m is = for an inverted image s″ is 4 for image to right of H″

m is 4 for an upright image s″ is = for image to left of H″

m is = for an inverted image

m is 4 for an upright image

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Example 2: Object inside Focal Point

The same object is placed 30 mm left of the left principal point of thesame lens. Where is the image formed, and what is the magnification?(See figure 1.3.)

or virtual image is 2.5 mm high and upright.

In this case, the lens is being used as a magnifier, and the image can beviewed only back through the lens.


Fundamental Optics


Example 3: Object at Focal Point

A 1-mm-high object is placed on the optical axis, 50 mm left of the firstprincipal point of an LDK-50.0-52.2-C (f =450 mm). Where is the imageformed, and what is the magnification? (See figure 1.4.)

or virtual image is 0.5 mm high and upright.

Typically, the first step in optical problem solving is to select a system focallength based on constraints such as magnification or conjugate distances(object and image distance). The relationship among focal length, objectposition, and image position is given by

This formula is referenced to figure 1.1 and the sign conventions givenin Sign Conventions.

By definition, magnification is the ratio of image size to object size or

This relationship can be used to recast the first formula into the followingforms:

where (s=s″) is the approximate object-to-image distance.

With a real lens of finite thickness, the image distance, object distance, andfocal length are all referenced to the principal points, not to the physicalcenter of the lens. By neglecting the distance between the lens’ principal points,known as the hiatus, s=s″ becomes the object-to-image distance. This sim-plification, called the thin-lens approximation, can speed up calculationwhen dealing with simple optical systems.

Example 1: Object outside Focal Point

A 1-mm-high object is placed on the optical axis, 200 mm left of the leftprincipal point of a LDX-25.0-51.0-C (f = 50 mm). Where is the imageformed, and what is the magnification? (See figure 1.2.)

or real image is 0.33 mm high and inverted.

object

F1

F2image

200 66.7

Figure 1.2 Example 1 (f = 50 mm, s = 200 mm, s″ = 66.7 mm)

objectF1 F2

image

Figure 1.3 Example 2 (f = 50 mm, s = 30 mm, s″= 475 mm)

1 1s

1s″

= +f

1 150

130

7575

302 5

″= −

″ = −

= ″ =−

= −

s

s

ms

s

mm

.

1 150

150

2525

500 5

″=

−−

″ = −

= ″ =−

= −

s

s

ms

s

mm

.

ms

s

h

h= ″ = ″

.

f ms s

m

fsm

m

fs s

mm

s m s s

=+ ″+

=+

=+ ″

+ +

+ = + ″

( )

( )

( )

1

1

2 1

1

2

(1.1)

(1.2)

(1.3)

(1.4)

(1.5)

(1.6)

s f

1 1 1

1 150

1200

66 7

66

″= −

″= −

″ =

= ″ =

s

s

s

ms

s

.

.

mm

77200

0 33 = .




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A simple graphical method can also be used to determine paraxial imagelocation and magnification. This graphical approach relies on two simpleproperties of an optical system. First, a ray that enters the system parallelto the optical axis crosses the optical axis at the focal point. Second, a raythat enters the first principal point of the system exits the system from thesecond principal point parallel to its original direction (i.e., its exit anglewith the optical axis is the same as its entrance angle). This method hasbeen applied to the three previous examples illustrated in figures 1.2through 1.4. Note that by using the thin-lens approximation, this secondproperty reduces to the statement that a ray passing through the centerof the lens is undeviated.

F-NUMBER AND NUMERICAL APERTURE

The paraxial calculations used to determine the necessary elementdiameter are based on the concepts of focal ratio (f-number or f/#) andnumerical aperture (NA). The f-number is the ratio of the focal length of thelens to its “effective” diameter, the clear aperture (CA).

To visualize the f-number, consider a lens with a positive focal lengthilluminated uniformly with collimated light. The f-number defines the angleof the cone of light leaving the lens which ultimately forms the image. Thisis an important concept when the throughput or light-gathering powerof an optical system is critical, such as when focusing light into a mono-chromator or projecting a high-power image.

The other term used commonly in defining this cone angle is numericalaperture. The NA is the sine of the angle made by the marginal ray withthe optical axis. By referring to figure 1.5 and using simple trigonometry,it can be seen that

and

Ray f-numbers can also be defined for any arbitrary ray if its conjugatedistance and the diameter at which it intersects the principal surface ofthe optical system are known.

NOTE

Because the sign convention given previously is not used universallyin all optics texts, the reader may notice differences in the paraxialformulas. However, results will be correct as long as a consistent setof formulas and sign conventions is used.



Fundamental Optics

object

F1F2 image

Figure 1.4 Example 3 (f = 450 mm, s = 50 mm, s″= 425 mm)

v

CA2

principal surface

f

Figure 1.5 F-number and numerical aperture

f-numberCA

=f .

NACA

= =sinv 2 f

NAf-number

=1

2( ).

(1.7)

(1.8)

(1.9)




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Example: System with Fixed Input NA

Two very common applications of simple optics involve coupling light intoan optical fiber or into the entrance slit of a monochromator. Although theseproblems appear to be quite different, they both have the same limitation— they have a fixed NA. For monochromators, this limit is usually expressedin terms of the f-number. In addition to the fixed NA, they both have a fixedentrance pupil (image) size.

Suppose it is necessary, using a singlet lens, to couple the output of anincandescent bulb with a filament 1 mm in diameter into an optical fiberas shown in figure 1.7. Assume that the fiber has a core diameter of100 mm and an NA of 0.25, and that the design requires that the totaldistance from the source to the fiber be 110 mm. Which lenses areappropriate?

By definition, the magnification must be 0.1. Letting s=s″ total 110 mm(using the thin-lens approximation), we can use equation 1.3,

to determine that the focal length is 9.1 mm. To determine the conjugatedistances, s and s″, we utilize equation 1.6,

and find that s = 100 mm and s″ = 10 mm.

We can now use the relationship NA = CA/2s or NA″ = CA/2s″ to deriveCA, the optimum clear aperture (effective diameter) of the lens.

With an image NA of 0.25 and an image distance (s″) of 10 mm,

and

Accomplishing this imaging task with a single lens therefore requires anoptic with a 9.1-mm focal length and a 5-mm diameter. Using a largerdiameter lens will not result in any greater system throughput because ofthe limited input NA of the optical fiber. The singlet lenses in this catalogthat meet these criteria are LPX-5.0-5.2-C, which is plano-convex, andLDX-6.0-7.7-C and LDX-5.0-9.9-C, which are biconvex.

Making some simple calculations has reduced our choice of lenses to justthree. The following chapter, Gaussian Beam Optics, discusses how tomake a final choice of lenses based on various performance criteria.

Imaging Properties of Lens Systemswww.cvimellesgriot .com

Fundamental Optics


THE OPTICAL INVARIANT

To understand the importance of the NA, consider its relation to magnification.Referring to figure 1.6,

The magnification of the system is therefore equal to the ratio of the NAson the object and image sides of the system. This powerful and useful resultis completely independent of the specifics of the optical system, and it canoften be used to determine the optimum lens diameter in situationsinvolving aperture constraints.

When a lens or optical system is used to create an image of a source, it isnatural to assume that, by increasing the diameter (f) of the lens, therebyincreasing its CA, we will be able to collect more light and thereby producea brighter image. However, because of the relationship between magnifi-cation and NA, there can be a theoretical limit beyond which increasing thediameter has no effect on light-collection efficiency or image brightness.

Since the NA of a ray is given by CA/2s, once a focal length and magni-fication have been selected, the value of NA sets the value of CA. Thus,if one is dealing with a system in which the NA is constrained on eitherthe object or image side, increasing the lens diameter beyond this valuewill increase system size and cost but will not improve performance (i.e.,throughput or image brightness). This concept is sometimes referred toas the optical invariant.

SAMPLE CALCULATION

To understand how to use this relationship between magnification and NA,consider the following example.

NA (object side)CA

NA″ (image side)CA

= =

= ″ =″

sin

sin

v

v

2

2

s

s

wwhich can be rearranged to show

CA

and

CA

=

= ″ ″

2

2

s

s

sin

sin

v

″ =″

=″

″

v

v

v

leading to

NANA

Since is simply t

s

s

s

s

sinsin

.

hhe magnification of the system,

we arrive at

NANA

m =″

.

(1.10)

(1.11)

(1.12)

(1.13)

(1.14)

(1.15)

f ms s

m=

+ ″+

(,

1 2)

)

((see eq. 1.3)

s m s s( ) ,+ = + ″1 (see eq. 1.6)

0 2520

. =CA

5 .=CA mm




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Fundamental Optics

v″

CA2

s″

CA

s

v

object side

image side

magnification = = = 0.1!h″h

0.11.0

filamenth = 1 mm

s + s″ = 110 mm

s = 100 mm s″ = 10 mm

fiber coreh″ = 0.1 mm

optical systemf = 9.1 mm

NA = = 0.025CA2s

CA = 5 mm

NA″ = = 0.25CA2s″

Figure 1.6 Numerical aperture and magnification

Figure 1.7 Optical system geometry for focusing the output of an incandescent bulb into an optical fiber




Many optical tasks require several lenses in order to achieve an acceptablelevel of performance. One possible approach to lens combinations is to con-sider each image formed by each lens as the object for the next lens and soon. This is a valid approach, but it is time consuming and unnecessary.

It is much simpler to calculate the effective (combined) focal length andprincipal-point locations and then use these results in any subsequentparaxial calculations (see figure 1.8). They can even be used in the opticalinvariant calculations described in the preceding section.

EFFECTIVE FOCAL LENGTH

The following formulas show how to calculate the effective focal lengthand principal-point locations for a combination of any two arbitrary com-ponents. The approach for more than two lenses is very simple: Calculatethe values for the first two elements, then perform the same calculation forthis combination with the next lens. This is continued until all lenses in thesystem are accounted for.

The expression for the combination focal length is the same whether lensseparation distances are large or small and whether f1 and f2 are positiveor negative:

This may be more familiar in the form

Notice that the formula is symmetric with respect to the interchange of thelenses (end-for-end rotation of the combination) at constant d. The nexttwo formulas are not.

COMBINATION FOCAL-POINT LOCATION

For all values of f1, f2, and d, the location of the focal point of the combinedsystem (s2″), measured from the secondary principal point of the secondlens (H2″), is given by

This can be shown by setting s1=d4f1 (see figure 1.8a), and solving

for s2″.

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Fundamental Optics


ff f

f f d=

+ −1 2

1 2. (1.16)

1 1 1

1 2 1 2f f f

d

f f= + − . (1.17)

sf f d

f f d2

2 1

1 2

″ =−

+ −( )

.

(1.18)

Symbols

fc = combination focal length (EFL), positive if combinationfinal focal point falls to the right of the combination secondaryprincipal point, negative otherwise (see figure 1.8c).

f1 = focal length of the first element (see figure 1.8a).

f2 = focal length of the second element.

d = distance from the secondary principal point of the firstelement to the primary principal point of the second element,positive if the primary principal point is to the right of thesecondary principal point, negative otherwise (see figure 1.8b).

s1″ = distance from the primary principal point of the firstelement to the final combination focal point (location of thefinal image for an object at infinity to the right of both lenses),positive if the focal point is to left of the first element’sprimary principal point (see figure 1.8d).

s2″ = distance from the secondary principal point of thesecond element to the final combination focal point (locationof the final image for an object at infinity to the left of bothlenses), positive if the focal point is to the right of the secondelement’s secondary principal point (see figure 1.8b).

zH = distance to the combination primary principal pointmeasured from the primary principal point of the first element,positive if the combination secondary principal point is tothe right of secondary principal point of second element(see figure 1.8d).

zH″ = distance to the combination secondary principal pointmeasured from the secondary principal point of the secondelement, positive if the combination secondary principal pointis to the right of the secondary principal point of the secondelement (see figure 1.8c).

Note: These paraxial formulas apply to coaxial combinationsof both thick and thin lenses immersed in air or any otherfluid with refractive index independent of position. Theyassume that light propagates from left to right through anoptical system.1 1 1

2 1 2f s s= +

″




COMBINATION SECONDARY PRINCIPAL-POINT LOCATION

Because the thin-lens approximation is obviously highly invalid for mostcombinations, the ability to determine the location of the secondary princi-pal point is vital for accurate determination of d when another elementis added. The simplest formula for this calculates the distance from thesecondary principal point of the final (second) element to the secondaryprincipal point of the combination (see figure 1.8b):

COMBINATION EXAMPLES

It is possible for a lens combination or system to exhibit principal planesthat are far removed from the system. When such systems are themselvescombined, negative values of d may occur. Probably the simplest exampleof a negative d-value situation is shown in figure 1.9. Meniscus lenses withsteep surfaces have external principal planes. When two of these lenses arebrought into contact, a negative value of d can occur. Other combined-lensexamples are shown in figures 1.10 through 1.13.

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Fundamental Optics

f1

H1

d

s1= d4 f1

H1″

fc

Hc

fc

H1 H1″

lens 1andlens 2

lens 1

H2 H2”

s2″

Hc″

zH″

lenscombination

lenscombination

zHzH

(a)

(b)

(c)

(d)

Figure 1.8 Lens combination focal length and principal planes

1 2 3 4

d>0

d<0

3 4 1 2

Figure 1.9 “Extreme” meniscus-form lenses withexternal principal planes (drawing not to scale)

z s f= ″ −2 . (1.19)




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Fundamental Optics


combinationsecondary

principal plane

focal plane

z

d

f<0f1 f2

s2″

Figure 1.10 Positive lenses separated by distance greaterthan ff1 == ff2: f is negative and both s2″ and z are positive. Lenssymmetry is not required.

combinationfocus

combinationsecondaryprincipal plane

f

z<0d s2″

Figure 1.12 Telephoto combination: The most importantcharacteristic of the telephoto lens is that the EFL, and hence the image size, can be made much larger than the distance fromthe first lens surface to the image would suggest by using a positive lens followed by a negative lens (but not necessarily thelens shapes shown in the figure). For example, f1 is positive andf2 = 4f1/2. Then f is negative for d less than f1/2, infinite for d = f1/2 (Galilean telescope or beam expander), and positive ford larger than f1/2. To make the example even more specific, catalog lenses LDX-50.8-130.4-C and LDK-42.0-52.2-C, with d = 78.2 mm, will yield s2″ = 2.0 m, f = 5.2 m, and z = 43.2 m.

H1″

f1

H2 H2″

d f2

Figure 1.11 Achromatic combinations: Air-spaced lenscombinations can be made nearly achromatic, even thoughboth elements are made from the same material. Achievingachromatism requires that, in the thin-lens approximation,

This is the basis for Huygens and Ramsden eyepieces.

This approximation is adequate for most thick-lens situations.The signs of f1, f2, and d are unrestricted, but d must have avalue that guarantees the existence of an air space. Elementshapes are unrestricted and can be chosen to compensate forother aberrations.

H

tc

ntc

n

s s″

H″

Figure 1.13 Condenser configuration: The convexvertices of a pair of identical plano-convex lenses areon contact. (The lenses could also be plano aspheres.) Becaused = 0, f = f1/2 = f2/2, f1/2 = s2″, and z = 0. The secondary princi-pal point of the second element and the secondary principalpoint of the combination coincide at H″, at depth tc /n beneaththe vertex of the plano surface of the second element, where tcis the element center thickness and n is the refractive index ofthe element. By symmetry, the primary principal point of thecombination is similarly located in the first element. Combina-tion conjugate distances must be measured from these points.

df f

=+( )

.1 2

2




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Fundamental Optics

After paraxial formulas have been used to select values for component focallength(s) and diameter(s), the final step is to select actual lenses. As in anyengineering problem, this selection process involves a number of tradeoffs,including performance, cost, weight, and environmental factors.

The performance of real optical systems is limited by several factors,including lens aberrations and light diffraction. The magnitude of theseeffects can be calculated with relative ease.

Numerous other factors, such as lens manufacturing tolerances andcomponent alignment, impact the performance of an optical system.Although these are not considered explicitly in the following discussion,it should be kept in mind that if calculations indicate that a lens systemonly just meets the desired performance criteria, in practice it may fall shortof this performance as a result of other factors. In critical applications,it is generally better to select a lens whose calculated performance issignificantly better than needed.

DIFFRACTION

Diffraction, a natural property of light arising from its wave nature, posesa fundamental limitation on any optical system. Diffraction is alwayspresent, although its effects may be masked if the system has significantaberrations. When an optical system is essentially free from aberrations,its performance is limited solely by diffraction, and it is referred to asdiffraction limited.

In calculating diffraction, we simply need to know the focal length(s) andaperture diameter(s); we do not consider other lens-related factors such asshape or index of refraction.

Since diffraction increases with increasing f-number, and aberrations decreasewith increasing f-number, determining optimum system performanceoften involves finding a point where the combination of these factorshas a minimum effect.

ABERRATIONS

To determine the precise performance of a lens system, we can trace thepath of light rays through it, using Snell’s law at each optical interfaceto determine the subsequent ray direction. This process, called ray tracing,is usually accomplished on a computer. When this process is completed,it is typically found that not all the rays pass through the points or posi-tions predicted by paraxial theory. These deviations from ideal imagingare called lens aberrations.

The direction of a light ray after refraction at the interface betweentwo homogeneous, isotropic media of differing index of refraction is givenby Snell’s law:

n1sinv1 = n2sinv2

where v1 is the angle of incidence, v2 is the angle of refraction, and both anglesare measured from the surface normal as shown in figure 1.14.

material 1index n 1

material 2index n 2

wavelength l

v1

v2

Figure 1.14 Refraction of light at a dielectric boundary

Technical Assistance

Detailed performance analysis of an optical system isaccomplished by using computerized ray-tracing software.CVI Melles Griot applications engineers are able to provide aray-tracing analysis of simple catalog-component systems. If youneed assistance in determining the performance of your opticalsystem, or in selecting optimum components for your particularapplication, please contact your nearest CVI Melles Griot office.

APPLICATION NOTE

(1.20)




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Even though tools for the precise analysis of an optical system are becom-ing easier to use and are readily available, it is still quite useful to have amethod for quickly estimating lens performance. This not only saves timein the initial stages of system specification, but can also help achieve abetter starting point for any further computer optimization.

The first step in developing these rough guidelines is to realize that the sinefunctions in Snell’s law can be expanded in an infinite Taylor series:

The first approximation we can make is to replace all the sine functionswith their arguments (i.e., replace sinv1 with v1 itself and so on). This iscalled first-order or paraxial theory because only the first terms of the sineexpansions are used. Design of any optical system generally starts withthis approximation using the paraxial formulas.

The assumption that sinv = v is reasonably valid for v close to zero (i.e.,high f-number lenses). With more highly curved surfaces (and particularlymarginal rays), paraxial theory yields increasingly large deviations fromreal performance because sinv ≠ v. These deviations are known as aber-rations. Because a perfect optical system (one without any aberrations)would form its image at the point and to the size indicated by paraxialtheory, aberrations are really a measure of how the image differs fromthe paraxial prediction.

As already stated, exact ray tracing is the only rigorous way to analyzereal lens surfaces. Before the advent of electronic computers, this wasexcessively tedious and time consuming. Seidel* addressed this issueby developing a method of calculating aberrations resulting from thev1

3/3! term. The resultant third-order lens aberrations are thereforecalled Seidel aberrations.

To simplify these calculations, Seidel put the aberrations of an opticalsystem into several different classifications. In monochromatic light theyare spherical aberration, astigmatism, field curvature, coma, and distor-tion. In polychromatic light there are also chromatic aberration and lat-eral color. Seidel developed methods to approximate each of theseaberrations without actually tracing large numbers of rays using all theterms in the sine expansions.

In actual practice, aberrations occur in combinations rather than alone. Thissystem of classifying them, which makes analysis much simpler, gives agood description of optical system image quality. In fact, even in the era ofpowerful ray-tracing software, Seidel’s formula for spherical aberration is stillwidely used.

SPHERICAL ABERRATION

Figure 1.15 illustrates how an aberration-free lens focuses incomingcollimated light. All rays pass through the focal point F″. The lower figureshows the situation more typically encountered in single lenses. The far-ther from the optical axis the ray enters the lens, the nearer to the lensit focuses (crosses the optical axis). The distance along the optical axisbetween the intercept of the rays that are nearly on the optical axis(paraxial rays) and the rays that go through the edge of the lens (marginalrays) is called longitudinal spherical aberration (LSA). The height atwhich these rays intercept the paraxial focal plane is called transversespherical aberration (TSA). These quantities are related by

TSA = LSA#tan(u″).

Spherical aberration is dependent on lens shape, orientation, andconjugate ratio, as well as on the index of refraction of the materialspresent. Parameters for choosing the best lens shape and orientationfor a given task are presented later in this chapter. However, the third-


Fundamental Optics


TSA

longitudinal spherical aberration

transverse spherical aberration

paraxial focal plane

F ″

F ″

u″

LSA

aberration-free lens

Figure 1.15 Spherical aberration of a plano-convex lens

sin / ! / ! / ! / ! . . .v v v v v v1 1 13

15

17

193 5 7 9= − + − + − (1.21)

(1.22)

* Ludwig von Seidel, 1857.




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order, monochromatic, spherical aberration of a plano-convex lens usedat infinite conjugate ratio can be estimated by

Theoretically, the simplest way to eliminate or reduce spherical aberrationis to make the lens surface(s) with a varying radius of curvature (i.e., anaspheric surface) designed to exactly compensate for the fact that sin v ≠ v

at larger angles. In practice, however, most lenses with high surfaceaccuracy are manufactured by grinding and polishing techniques thatnaturally produce spherical or cylindrical surfaces. The manufacture ofaspheric surfaces is more complex, and it is difficult to produce a lens ofsufficient surface accuracy to eliminate spherical aberration completely.Fortunately, these aberrations can be virtually eliminated, for a chosenset of conditions, by combining the effects of two or more spherical (orcylindrical) surfaces.

In general, simple positive lenses have undercorrected spherical aberration,and negative lenses usually have overcorrected spherical aberration. Bycombining a positive lens made from low-index glass with a negative lensmade from high-index glass, it is possible to produce a combination in whichthe spherical aberrations cancel but the focusing powers do not. Thesimplest examples of this are cemented doublets, such as the LAO serieswhich produce minimal spherical aberration when properly used.

ASTIGMATISM

When an off-axis object is focused by a spherical lens, the natural asym-metry leads to astigmatism. The system appears to have two differentfocal lengths.

As shown in figure 1.16, the plane containing both optical axis and objectpoint is called the tangential plane. Rays that lie in this plane are calledtangential, or meridional, rays. Rays not in this plane are referred to as skewrays. The chief, or principal, ray goes from the object point through thecenter of the aperture of the lens system. The plane perpendicular to thetangential plane that contains the principal ray is called the sagittal orradial plane.

The figure illustrates that tangential rays from the object come to a focus closerto the lens than do rays in the sagittal plane. When the image is evaluatedat the tangential conjugate, we see a line in the sagittal direction. A line inthe tangential direction is formed at the sagittal conjugate. Between theseconjugates, the image is either an elliptical or a circular blur. Astigmatismis defined as the separation of these conjugates.

The amount of astigmatism in a lens depends on lens shape only whenthere is an aperture in the system that is not in contact with the lens itself.(In all optical systems there is an aperture or stop, although in many casesit is simply the clear aperture of the lens element itself.) Astigmatism stronglydepends on the conjugate ratio.



Fundamental Optics

tangential image(focal line)

principal raytangential plane

optical axis

object point

optical system

sagittal plane

paraxialfocal plane

sagittal image (focal line)

Figure 1.16 Astigmatism represented by sectional views

spot size due to spherical aberration

f/#=

0 0673

..

f(1.23)




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COMA

In spherical lenses, different parts of the lens surface exhibit different degreesof magnification. This gives rise to an aberration known as coma. As shownin figure 1.17, each concentric zone of a lens forms a ring-shaped image calleda comatic circle. This causes blurring in the image plane (surface) of off-axisobject points. An off-axis object point is not a sharp image point, but itappears as a characteristic comet-like flare. Even if spherical aberration iscorrected and the lens brings all rays to a sharp focus on axis, a lens maystill exhibit coma off axis. See figure 1.18.

As with spherical aberration, correction can be achieved by using multiplesurfaces. Alternatively, a sharper image may be produced by judiciouslyplacing an aperture, or stop, in an optical system to eliminate the moremarginal rays.

FIELD CURVATURE

Even in the absence of astigmatism, there is a tendency of optical systemsto image better on curved surfaces than on flat planes. This effect is calledfield curvature (see figure 1.19). In the presence of astigmatism, thisproblem is compounded because two separate astigmatic focal surfacescorrespond to the tangential and sagittal conjugates.

Field curvature varies with the square of field angle or the square of imageheight. Therefore, by reducing the field angle by one-half, it is possible to reducethe blur from field curvature to a value of 0.25 of its original size.

Positive lens elements usually have inward curving fields, and negativelenses have outward curving fields. Field curvature can thus be corrected tosome extent by combining positive and negative lens elements.


Fundamental Optics


points on lens

1

1

2′03 3

4 2

42

1′

3′4′

1′2′

3′4′

correspondingpoints on S

60∞

1

24

3

1′

4′

3′

2′

S

1

11′

S

P,O

1

1′

1′

Figure 1.17 Imaging an off-axis point source by a lens with positive transverse coma

positive transverse coma

focal plane

Figure 1.18 Positive transverse coma

spherical focal surface

Figure 1.19 Field curvature




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DISTORTION

The image field not only may have curvature but may also be distorted.The image of an off-axis point may be formed at a location on this surfaceother than that predicted by the simple paraxial equations. This distor-tion is different from coma (where rays from an off-axis point fail tomeet perfectly in the image plane). Distortion means that even if a perfectoff-axis point image is formed, its location on the image plane is notcorrect. Furthermore, the amount of distortion usually increases withincreasing image height. The effect of this can be seen as two differentkinds of distortion: pincushion and barrel (see figure 1.20). Distortion doesnot lower system resolution; it simply means that the image shape doesnot correspond exactly to the shape of the object. Distortion is a sepa-ration of the actual image point from the paraxially predicted locationon the image plane and can be expressed either as an absolute value oras a percentage of the paraxial image height.

It should be apparent that a lens or lens system has opposite types ofdistortion depending on whether it is used forward or backward. Thismeans that if a lens were used to make a photograph, and then usedin reverse to project it, there would be no distortion in the final screenimage. Also, perfectly symmetrical optical systems at 1:1 magnificationhave no distortion or coma.



Fundamental Optics

BARRELDISTORTION

PINCUSHIONDISTORTION

OBJECT

Figure 1.20 Pincushion and barrel distortion

white light ray

longitudinalchromaticaberration

blue focal point

red focal point

red light ray

blue light ray

Figure 1.21 Longitudinal chromatic aberration

Aperture Field Angle Image Height

Aberration (f) (v) (y)

Lateral Spherical f3 — —

Longitudinal Spherical f2 — —

Coma f2 v y

Astigmatism f v2 y2

Field Curvature f v2 y2

Distortion — v3 y3

Chromatic — — —

Variations of Aberrations with Aperture,Field Angle, and Image Height

CHROMATIC ABERRATION

The aberrations previously described are purely a function of the shapeof the lens surfaces, and they can be observed with monochromatic light.Other aberrations, however, arise when these optics are used to trans-form light containing multiple wavelengths. The index of refraction of amaterial is a function of wavelength. Known as dispersion, this is discussedin Material Properties. From Snell’s law (see equation 1.20), it can beseen that light rays of different wavelengths or colors will be refractedat different angles since the index is not a constant. Figure 1.21 showsthe result when polychromatic collimated light is incident on a positivelens element. Because the index of refraction is higher for shorter wave-lengths, these are focused closer to the lens than the longer wavelengths.Longitudinal chromatic aberration is defined as the axial distance fromthe nearest to the farthest focal point. As in the case of spherical aberration,positive and negative elements have opposite signs of chromatic aberration.Once again, by combining elements of nearly opposite aberration to forma doublet, chromatic aberration can be partially corrected. It is neces-sary to use two glasses with different dispersion characteristics, so thatthe weaker negative element can balance the aberration of the stronger,positive element.




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LATERAL COLOR

Lateral color is the difference in image height between blue and red rays.Figure 1.22 shows the chief ray of an optical system consisting of a simplepositive lens and a separate aperture. Because of the change in index withwavelength, blue light is refracted more strongly than red light, whichis why rays intercept the image plane at different heights. Stated simply,magnification depends on color. Lateral color is very dependent on systemstop location.

For many optical systems, the third-order term is all that may be neededto quantify aberrations. However, in highly corrected systems or in thosehaving large apertures or a large angular field of view, third-order theoryis inadequate. In these cases, exact ray tracing is absolutely essential.


Fundamental Optics


aperture

red light ray lateral color

blue light ray

focal plane

Figure 1.22 Lateral Color

Achromatic Doublets Are Superior toSimple Lenses

Because achromatic doublets correct for spherical as well aschromatic aberration, they are often superior to simple lensesfor focusing collimated light or collimating point sources, even inpurely monochromatic light.

Although there is no simple formula that can be used toestimate the spot size of a doublet, the tables in Spot Size givesample values that can be used to estimate the performance ofcatalog achromatic doublets.

APPLICATION NOTE




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At infinite conjugate with a typical glass singlet, the plano-convex shape(q = 1), with convex side toward the infinite conjugate, performs nearlyas well as the best-form lens. Because a plano-convex lens costs muchless to manufacture than an asymmetric biconvex singlet, these lensesare quite popular. Furthermore, this lens shape exhibits near-minimumtotal transverse aberration and near-zero coma when used off axis,thus enhancing its utility.

For imaging at unit magnification (s = s″ = 2f), a similar analysis wouldshow that a symmetric biconvex lens is the best shape. Not only isspherical aberration minimized, but coma, distortion, and lateral chro-matic aberration exactly cancel each other out. These results are trueregardless of material index or wavelength, which explains the utilityof symmetric convex lenses, as well as symmetrical optical systems ingeneral. However, if a remote stop is present, these aberrations may notcancel each other quite as well.

For wide-field applications, the best-form shape is definitely not the optimumsinglet shape, especially at the infinite conjugate ratio, since it yieldsmaximum field curvature. The ideal shape is determined by the situationand may require rigorous ray-tracing analysis. It is possible to achieve muchbetter correction in an optical system by using more than one element.The cases of an infinite conjugate ratio system and a unit conjugate ratiosystem are discussed in the following section.

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Fundamental Optics

Aberrations described in the preceding section are highly dependent onapplication, lens shape, and material of the lens (or, more exactly, itsindex of refraction). The singlet shape that minimizes spherical aberrationat a given conjugate ratio is called best-form. The criterion for best-form atany conjugate ratio is that the marginal rays are equally refracted at eachof the lens/air interfaces. This minimizes the effect of sinv ≠ v. It is alsothe criterion for minimum surface-reflectance loss. Another benefit isthat absolute coma is nearly minimized for best-form shape, at both infiniteand unit conjugate ratios.

To further explore the dependence of aberrations on lens shape, it ishelpful to make use of the Coddington shape factor, q, defined as

Figure 1.23 shows the transverse and longitudinal spherical aberrationsof a singlet lens as a function of the shape factor, q. In this particularinstance, the lens has a focal length of 100 mm, operates at f/5, has anindex of refraction of 1.518722 (BK7 at the mercury green line, 546.1 nm),and is being operated at the infinite conjugate ratio. It is also assumedthat the lens itself is the aperture stop. An asymmetric shape that corre-sponds to a q-value of about 0.7426 for this material and wavelength isthe best singlet shape for on-axis imaging. It is important to note that thebest-form shape is dependent on refractive index. For example, with ahigh-index material, such as silicon, the best-form lens for the infiniteconjugate ratio is a meniscus shape.

SHAPE FACTOR (q)

42

AB

ERR

ATI

ON

S IN

MIL

LIM

ETER

S

41.5 41 40.5 0 0.5 1 1.5 2

5

4

3

2

1

exact transverse sphericalaberration (TSA)

exact longitudinal spherical aberration (LSA)

Figure 1.23 Aberrations of positive singlets at infinite conjugate ratio as a function of shape

qr r

r r=

+−

( )( )

.2 1

2 1

(1.24)




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Previous examples indicate that an achromat is superior in performanceto a singlet when used at the infinite conjugate ratio and at low f-numbers.Since the unit conjugate case can be thought of as two lenses, eachworking at the infinite conjugate ratio, the next step is to replace theplano-convex singlets with achromats, yielding a four-element system. Thethird part of figure 1.25 shows a system composed of two LAO-40.0-18.0lenses. Once again, spherical aberration is not evident, even in the f/2.7 ray.

Lens Combinationswww.cvimellesgriot .com

Fundamental Optics


INFINITE CONJUGATE RATIO

As shown in the previous discussion, the best-form singlet lens for useat infinite conjugate ratios is generally nearly plano-convex. Figure 1.24shows a plano-convex lens (LPX-15.0-10.9-C) with incoming collimatedlight at a wavelength of 546.1 nm. This drawing, including the rays tracedthrough it, is shown to exact scale. The marginal ray (ray f-number 1.5)strikes the paraxial focal plane significantly off the optical axis.

This situation can be improved by using a two-element system. The secondpart of the figure shows a precision achromat (LAO-21.0-14.0), whichconsists of a positive low-index (crown glass) element cemented to anegative meniscus high-index (flint glass) element. This is drawn to thesame scale as the plano-convex lens. No spherical aberration can be dis-cerned in the lens. Of course, not all of the rays pass exactly through theparaxial focal point; however, in this case, the departure is measuredin micrometers, rather than in millimeters, as in the case of the plano-convex lens. Additionally, chromatic aberration (not shown) is much bettercorrected in the doublet. Even though these lenses are known as achromaticdoublets, it is important to remember that even with monochromatic lightthe doublet’s performance is superior.

Figure 1.24 also shows the f-number at which singlet performance becomesunacceptable. The ray with f-number 7.5 practically intercepts the paraxialfocal point, and the f/3.8 ray is fairly close. This useful drawing, which canbe scaled to fit a plano-convex lens of any focal length, can be used toestimate the magnitude of its spherical aberration, although lens thicknessaffects results slightly.

UNIT CONJUGATE RATIO

Figure 1.25 shows three possible systems for use at the unit conjugate ratio.All are shown to the same scale and using the same ray f-numbers with alight wavelength of 546.1 nm. The first system is a symmetric biconvexlens (LDX-21.0-19.2-C), the best-form singlet in this application. Clearly,significant spherical aberration is present in this lens at f/2.7. Not untilf/13.3 does the ray closely approach the paraxial focus.

A dramatic improvement in performance is gained by using two identicalplano-convex lenses with convex surfaces facing and nearly in contact.Those shown in figure 1.25 are both LPX-20.0-20.7-C. The combinationof these two lenses yields almost exactly the same focal length as thebiconvex lens. To understand why this configuration improves performanceso dramatically, consider that if the biconvex lens were split down themiddle, we would have two identical plano-convex lenses, each working atan infinite conjugate ratio, but with the convex surface toward the focus. Thisorientation is opposite to that shown to be optimum for this shape lens. Onthe other hand, if these lenses are reversed, we have the system just describedbut with a better correction of the spherical aberration.

PLANO-CONVEX LENS

ray f-numbers

1.92.53.87.5

1.5

1.92.53.87.5

1.5

ACHROMAT

paraxial image plane

LPX-15.0-10.9-C

LAO-21.0-14.0

Figure 1.24 Single-element plano-convex lens comparedwith a two-element achromat




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Fundamental Optics

SYMMETRIC BICONVEX LENS

ray f-numbers

IDENTICAL PLANO-CONVEX LENSES

IDENTICAL ACHROMATS

2.73.34.46.7

13.3

2.73.34.46.7

13.3

2.73.34.46.7

13.3

paraxial image plane

LDX-21.0-19.2-C

LPX-20.0-20.7-C

LAO-40.0-18.0

Figure 1.25 Three possible systems for use at the unit conjugate ratio




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CIRCULAR APERTURE

Fraunhofer diffraction at a circular aperture dictates the fundamentallimits of performance for circular lenses. It is important to remember thatthe spot size, caused by diffraction, of a circular lens is

d = 2.44l(f/#)

where d is the diameter of the focused spot produced from plane-waveillumination and l is the wavelength of light being focused. Notice thatit is the f-number of the lens, not its absolute diameter, that determinesthis limiting spot size.

The diffraction pattern resulting from a uniformly illuminated circularaperture actually consists of a central bright region, known as the Airydisc (see figure 1.27), which is surrounded by a number of much fainterrings. Each ring is separated by a circle of zero intensity. The irradiancedistribution in this pattern can be described by

where

Diffraction Effectswww.cvimellesgriot .com

Fundamental Optics


In all light beams, some energy is spread outside the region predicted bygeometric propagation. This effect, known as diffraction, is a fundamentaland inescapable physical phenomenon. Diffraction can be understood byconsidering the wave nature of light. Huygens’ principle (figure 1.26)states that each point on a propagating wavefront is an emitter of secondarywavelets. The propagating wave is then the envelope of these expandingwavelets. Interference between the secondary wavelets gives rise to afringe pattern that rapidly decreases in intensity with increasing anglefrom the initial direction of propagation. Huygens’ principle nicelydescribes diffraction, but rigorous explanation demands a detailed studyof wave theory.

Diffraction effects are traditionally classified into either Fresnel or Fraun-hofer types. Fresnel diffraction is primarily concerned with what happens tolight in the immediate neighborhood of a diffracting object or aperture. Itis thus only of concern when the illumination source is close to this apertureor object, known as the near field. Consequently, Fresnel diffraction is rarelyimportant in most classical optical setups, but it becomes very important insuch applications as digital optics, fiber optics, and near-field microscopy.

Fraunhofer diffraction, however, is often important even in simple opticalsystems. This is the light-spreading effect of an aperture when the aperture(or object) is illuminated with an infinite source (plane-wave illumination)and the light is sensed at an infinite distance (far-field) from this aperture.

From these overly simple definitions, one might assume that Fraunhoferdiffraction is important only in optical systems with infinite conjugate,whereas Fresnel diffraction equations should be considered at finite conju-gate ratios. Not so. A lens or lens system of finite positive focal length withplane-wave input maps the far-field diffraction pattern of its aperture ontothe focal plane; therefore, it is Fraunhofer diffraction that determines thelimiting performance of optical systems. More generally, at any conjugateratio, far-field angles are transformed into spatial displacements in theimage plane.

aperture

secondarywavelets

wavefront wavefront

some light diffractedinto this region

Figure 1.26 Huygens’ principle

AIRY DISC DIAMETER = 2.44 l f/#

Figure 1.27 Center of a typical diffraction pattern for acircular aperture

I IJ x

xx = 0

122 ( )⎧

⎩⎪⎫⎭⎪

(1.26)

I

J x

0

1

=

( ) =peak irradiance in the image

Bessel function of thee first kind of order unity

= −( )−( )=

∞ + −

∑xx

n nn

n n

n1

1 21

1 2 2

2! ! −−1

(1.25)




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and

where

l = wavelength

D = aperture diameter

v = angular radius from the pattern maximum.

This useful formula shows the far-field irradiance distribution from a uniformlyilluminated circular aperture of diameter D.

SLIT APERTURE

A slit aperture, which is mathematically simpler, is useful in relation tocylindrical optical elements. The irradiance distribution in the diffractionpattern of a uniformly illuminated slit aperture is described by

ENERGY DISTRIBUTION TABLE

The accompanying table shows the major features of pure (unaberrated)Fraunhofer diffraction patterns of circular and slit apertures. The tableshows the position, relative intensity, and percentage of total patternenergy corresponding to each ring or band. It is especially convenient tocharacterize positions in either pattern with the same variable x. Thisvariable is related to field angle in the circular aperture case by

where D is the aperture diameter. For a slit aperture, this relationship isgiven by

where w is the slit width, p has its usual meaning, and D, w, and l areall in the same units (preferably millimeters). Linear instead of angularfield positions are simply found from r=s″tanv where s″ is the secondaryconjugate distance. This last result is often seen in a different form, namelythe diffraction-limited spot-size equation, which, for a circular lens is

This value represents the smallest spot size that can be achieved by anoptical system with a circular aperture of a given f-number, and it is thediameter of the first dark ring, where the intensity has dropped to zero.

The graph in figure 1.28 shows the form of both circular and slit aperturediffraction patterns when plotted on the same normalized scale. Aperturediameter is equal to slit width so that patterns between x-values andangular deviations in the far-field are the same.

GAUSSIAN BEAMS

Apodization, or nonuniformity of aperture irradiance, alters diffractionpatterns. If pupil irradiance is nonuniform, the formulas and results givenpreviously do not apply. This is important to remember because mostlaser-based optical systems do not have uniform pupil irradiance. The out-put beam of a laser operating in the TEM00 mode has a smooth Gaussianirradiance profile. Formulas used to determine the focused spot size fromsuch a beam are discussed in Gaussian Beam Optics. Furthermore, whendealing with Gaussian beams, the location of the focused spot also departsfrom that predicted by the paraxial equations given in this chapter. Thisis also detailed in Gaussian Beam Optics.



Fundamental Optics

Rayleigh Criterion

In imaging applications, spatial resolution is ultimately limitedby diffraction. Calculating the maximum possible spatialresolution of an optical system requires an arbitrary definitionof what is meant by resolving two features. In the Rayleighcriterion, it is assumed that two separate point sources can beresolved when the center of the Airy disc from one overlaps thefirst dark ring in the diffraction pattern of the second. In thiscase, the smallest resolvable distance, d, is

APPLICATION NOTE

d l= =0.61NA

1.22 f/#l

( )

I Ix

xx = ⎧

⎩⎪⎫⎭⎪0

2sin(1.27)

I

xw

w

0 =

=

=

=

peak irradiance in imagewhere

wavelengthwhere

slit w

p v

l

l

sin

iidth

angular deviation from pattern maximum.v =

sinvl

p=

x

D(1.28)

sinvl

p=

x

w(1.29)

d = ( )2 44. /# fl (see eq. 1.25)

xD

=p

lvsin




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Fundamental Optics


SLIT APERTURE

90.3% incentral maximum

95.0% within the twoadjoining subsidiary maxima

POSITION IN IMAGE PLANE (x)

CIRCULAR APERTURE

91.0% within first bright ring

83.9% in Airy disc

46 45 44 43 42 414748 0 1 2 3 4 5 6 7 80.0

.1

.2

.3

.4

.5

.6

.7

.8

.91.0

slitaperture

circularaperture

NO

RM

ALI

ZED

PA

TTER

N IR

RA

DIA

NC

E (y

)

Figure 1.28 Fraunhofer diffraction pattern of a singlet slit superimposed on the Fraunhofer diffraction patternof a circular aperture

Relative Energy Relative Energy

Position Intensity in Ring Position Intensity in Band

Ring or Band (x) (Ix/I0) (%) (x) (Ix/I0) (%)

Central Maximum 0.0 1.0 83.8 0.0 1.0 90.3

First Dark 1.22p 0.0 1.00p 0.0

First Bright 1.64p 0.0175 7.2 1.43p 0.0472 4.7

Second Dark 2.23p 0.0 2.00p 0.0

Second Bright 2.68p 0.0042 2.8 2.46p 0.0165 1.7

Third Dark 3.24p 0.0 3.00p 0.0

Third Bright 3.70p 0.0016 1.5 3.47p 0.0083 0.8

Fourth Dark 4.24p 0.0 4.00p 0.0

Fourth Bright 4.71p 0.0008 1.0 4.48p 0.0050 0.5

Fifth Dark 5.24p 0.0 5.00p 0.0

Slit Aperture Circular Aperture

Energy Distribution in the Diffraction Pattern of a Circular or Slit Aperture

Note: Position variable (x) is defined in the text.




limited, it is pointless to strive for better resolution. This level of resolutioncan be achieved easily with a plano-convex lens.

While angular divergence decreases with increasing focal length, sphericalaberration of a plano-convex lens increases with increasing focal length.To determine the appropriate focal length, set the spherical aberrationformula for a plano-convex lens equal to the source (spot) size:

This ensures a lens that meets the minimum performance needed. Toselect a focal length, make an arbitrary f-number choice. As can be seenfrom the relationship, as we lower the f-number (increase collectionefficiency), we decrease the focal length, which will worsen the resultantdivergence angle (minimum divergence = 1 mm/f).

In this example, we will accept f/2 collection efficiency, which gives usa focal length of about 120 mm. For f/2 operation we would need a min-imum diameter of 60 mm. The LPX-60.0-62.2-C fits this specificationexactly. Beam divergence would be about 8 mrad.

Finally, we need to verify that we are not operating below the theoreticaldiffraction limit. In this example, the numbers (1-mm spot size) indicatethat we are not, since

diffraction-limited spot size = 2.44#0.5 mm#2 = 2.44 mm.

Example 2: Coupling an Incandescent Source into a Fiber

In Imaging Properties of Lens Systems we considered a system in which theoutput of an incandescent bulb with a filament of 1 mm in diameter wasto be coupled into an optical fiber with a core diameter of 100 µm and anumerical aperture of 0.25. From the optical invariant and other constraintsgiven in the problem, we determined that f = 9.1 mm, CA = 5 mm, s = 100mm, s″ = 10 mm, NA″ = 0.25, and NA = 0.025 (or f/2 and f/20). The singletlenses that match these specifications are the plano-convex LPX-5.0-5.2-Cor biconvex lenses LDX-6.0-7.7-C and LDX-5.0-9.9-C. The closest achromatwould be the LAO-10.0-6.0.

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Lens Selectionwww.cvimellesgriot .com


Fundamental Optics

Having discussed the most important factors that affect the performanceof a lens or a lens system, we will now address the practical matter ofselecting the optimum catalog components for a particular task. The fol-lowing useful relationships are important to keep in mind throughoutthe selection process:

$ Diffraction-limited spot size = 2.44 l f/#

$ Approximate on-axis spot size of a plano-convex lens at the infinite conjugate resulting from spherical aberration =

$ Optical invariant =

Example 1: Collimating an Incandescent Source

Produce a collimated beam from a quartz halogen bulb having a 1-mm-square filament. Collect the maximum amount of light possible and producea beam with the lowest possible divergence angle.

This problem, illustrated in figure 1.29, involves the typical tradeoffbetween light-collection efficiency and resolution (where a beam is beingcollimated rather than focused, resolution is defined by beam diver-gence). To collect more light, it is necessary to work at a low f-number,but because of aberrations, higher resolution (lower divergence angle)will be achieved by working at a higher f-number.

In terms of resolution, the first thing to realize is that the minimumdivergence angle (in radians) that can be achieved using any lens systemis the source size divided by system focal length. An off-axis ray (fromthe edge of the source) entering the first principal point of the systemexits the second principal point at the same angle. Therefore, increasingthe system focal length improves this limiting divergence because thesource appears smaller.

An optic that can produce a spot size of 1 mm when focusing a perfectlycollimated beam is therefore required. Since source size is inherently

f

v min

v min = source size f

Figure 1.29 Collimating an incandescent source

0 067 13.

/#.

f mm

f=

0 0673

.

/#

f

f

m =″

NANA

.




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We can immediately reject the biconvex lenses because of spherical aberration. We can estimate the performance of the LPX-5.0-5.2-Con the focusing side by using our spherical aberration formula:

We will ignore, for the moment, that we are not working at the infiniteconjugate.

This is slightly smaller than the 100-µm spot size we are trying to achieve.However, since we are not working at infinite conjugate, the spot size willbe larger than that given by our simple calculation. This lens is thereforelikely to be marginal in this situation, especially if we consider chromaticaberration. A better choice is the achromat. Although a computer raytrace would be required to determine its exact performance, it is virtuallycertain to provide adequate performance.

Example 3: Symmetric Fiber-to-Fiber Coupling

Couple an optical fiber with an 8-µm core and a 0.15 numerical apertureinto another fiber with the same characteristics. Assume a wavelength of0.5 µm.

This problem, illustrated in figure 1.30, is essentially a 1:1 imaging situation.We want to collect and focus at a numerical aperture of 0.15 or f/3.3, andwe need a lens with an 8-µm spot size at this f-number. Based on the lenscombination discussion in Lens Combination Formulas, our most likelysetup is either a pair of identical plano-convex lenses or achromats, faced

front to front. To determine the necessary focal length for a plano-convexlens, we again use the spherical aberration estimate formula:

This formula yields a focal length of 4.3 mm and a minimum diameter of1.3 mm. The LPX-4.2-2.3-BAK1 meets these criteria. The biggest problemwith utilizing these tiny, short focal length lenses is the practical consid-erations of handling, mounting, and positioning them. Because using apair of longer focal length singlets would result in unacceptable perfor-mance, the next step might be to use a pair of the slightly longer focallength, larger achromats, such as the LAO-10.0-6.0. The performance data,given in Spot Size, show that this combination does provide the required8-mm spot diameter.

Because fairly small spot sizes are being considered here, it is importantto make sure that the system is not being asked to work below the diffractionlimit:

Since this is half the spot size caused by aberrations, it can be safelyassumed that diffraction will not play a significant role here.

An entirely different approach to a fiber-coupling task such as this wouldbe to use a pair of spherical ball lenses (LMS-LSFN series) or one of thegradient-index lenses (LGT series).


Fundamental Optics


s = f s″= f

Figure 1.30 Symmetric fiber-to-fiber coupling

spot size

m.= =0 067 10

2843

. ( )m

0 0673 3

0 0083.

.. .

mm

f=

. .2 44 0 5 3 3 4. .= m mm m × ×




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Example 4: Diffraction-Limited Performance

Determine at what f-number a plano-convex lens being used at an infiniteconjugate ratio with 0.5-mm wavelength light becomes diffraction lim-ited (i.e., the effects of diffraction exceed those caused by aberration).

To solve this problem, set the equations for diffraction-limited spot sizeand third-order spherical aberration equal to each other. The resultdepends upon focal length, since aberrations scale with focal length,while diffraction is solely dependent upon f-number. By substitutingsome common focal lengths into this formula, we get f/8.6 at f = 100 mm,f/7.2 at f = 50 mm, and f/4.8 at f = 10 mm.

When working with these focal lengths (and under the conditions previouslystated), we can assume essentially diffraction-limited performance abovethese f-numbers. Keep in mind, however, that this treatment does not takeinto account manufacturing tolerances or chromatic aberration, which willbe present in polychromatic applications.



Fundamental Optics

2 44 0 5 0 067

54 9

3

1 4

. . /#.

/#

/# ( . ) ./

× × =×

= ×

m ff

or

f

mf

f

Spherical Ball Lenses for Fiber Coupling

Spheres are arranged so that the fiber end is located at the focalpoint. The output from the first sphere is then collimated. If twospheres are aligned axially to each other, the beam will betransferred from one focal point to the other. Translationalalignment sensitivity can be reduced by enlarging the beam.Slight negative defocusing of the ball can reduce the sphericalaberration third-order contribution common to all couplingsystems. Additional information can be found in “Lens Couplingin Fiber Optic Devices: Efficiency Limits,” by A. Nicia, AppliedOptics, vol. 20, no. 18, pp 3136–45, 1981. Off-axis aberrationsare absent since the fiber diameters are so much smaller thanthe coupler focal length.

APPLICATION NOTE

opticalfiber

collimatedlight section

opticalfiber

LMS-LSFN coupling sphere

LMS-LSFN coupling sphere

uncoated narrow band

fbfb




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The effect on spot size caused by spherical aberration is strongly dependenton f-number. For a plano-convex singlet, spherical aberration is inverselydependent on the cube of the f-number. For doublets, this relationship canbe even higher. On the other hand, the spot size caused by diffractionincreases linearly with f-number. Thus, for some lens types, spot size atfirst decreases and then increases with f-number, meaning that there issome optimum performance point at which both aberrations and diffractioncombine to form a minimum.

Unfortunately, these results cannot be generalized to situations in whichthe lenses are used off axis. This is particularly true of the achromat/apla-natic meniscus lens combinations because their performance degradesrapidly off axis.

Spot Sizewww.cvimellesgriot .com

Fundamental Optics


In general, the performance of a lens or lens system in a specific circumstanceshould be determined by an exact trigonometric ray trace. CVI Melles Griotapplications engineers can supply ray-tracing data for particular lensesand systems of catalog components on request. In certain situations,however, some simple guidelines can be used for lens selection. The opti-mum working conditions for some of the lenses in this catalog havealready been presented. The following tables give some quantitativeresults for a variety of simple and compound lens systems, which can beconstructed from standard catalog optics.

In interpreting these tables, remember that these theoretical valuesobtained from computer ray tracing consider only the effects of idealgeometric optics. Effects of manufacturing tolerances have not been con-sidered. Furthermore, remember that using more than one elementprovides a higher degree of correction but makes alignment more difficult.When actually choosing a lens or a lens system, it is important to notethe tolerances and specifications clearly described for each CVI Melles Griotlens in the product listings.

The tables give the diameter of the spot for a variety of lenses used atseveral different f-numbers. All the tables are for on-axis, uniformlyilluminated, collimated input light at 546.1 nm. They assume that thelens is facing in the direction that produces a minimum spot size. Whenthe spot size caused by aberrations is smaller or equal to the diffraction-limited spot size, the notation “DL” appears next to the entry. The shorterfocal length lenses produce smaller spot sizes because aberrationsincrease linearly as a lens is scaled up.

f/# LDX-5.0-9.9-C LPX-8.0-5.2-C LAO-10.0-6.0

f/2 — 94 11

f/3 36 25 7

f/5 8 6.7 (DL) 6.7 (DL)

f/10 13.3 (DL) 13.3 (DL) 13.3 (DL)

Focal Length = 10 mm

*Diffraction-limited performance is indicated by DL.

f/# LDX-50.0-60.0-C LPX-30.0-31.1-C LAO-60.0-30.0 LAO-100.0-31.5 & MENP-31.5-6.0-146.4-NSF8

f/2 816 600 — —

f/3 217 160 34 4 (DL)

f/5 45 33 10 6.7 (DL)

f/10 13.3 (DL) 13.3 (DL) 13.3 (DL) 13.3 (DL)

Spot Size (mm)*

Spot Size (mm)*



LAO-50.0-18.0 &

f/# LPX-18.5-15.6-C LAO-30.0-12.5 MENP-18.0-4.0-73.5-NSF8

f/2 295 — 3

f/3 79 7 4 (DL)

f/5 17 6.7 (DL) 6.9 (DL)

f/10 13.3 (DL) 13.3 (DL) 13.8 (DL)



Spot Size (mm)*




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If a plano-convex lens of focal length f1 oriented in the normal direction iscombined with a plano-concave lens of focal length f2 oriented in its reversedirection, the total spherical aberration of the system is

After setting this equation to zero, we obtain

To make the magnitude of aberration contributions of the two elementsequal so they will cancel out, and thus correct the system, select the focallength of the positive element to be 3.93 times that of the negativeelement.

Figure 1.32(a) shows a beam-expander system made up of catalog elements,in which the focal length ratio is 4:1. This simple system is corrected toabout 1/6 wavelength at 632.8 nm, even though the objective is operatingat f/4 with a 20-mm aperture diameter. This is remarkably good wavefrontcorrection for such a simple system; one would normally assume that adoublet objective would be needed and a complex diverging lens as well.This analysis does not take into account manufacturing tolerances.

A beam expander of lower magnification can also be derived from thisinformation. If a symmetric-convex objective is used together] with areversed plano-concave diverging lens, the aberration coefficients are inthe ratio of 1.069/0.403 = 2.65. Figure 1.32(b) shows a system of cataloglenses that provides a magnification of 2.7 (the closest possible given theavailable focal lengths). The maximum wavefront error in this case is onlya quarter-wave, even though the objective is working at f/3.3.

Aberration Balancingwww.cvimellesgriot .com


Fundamental Optics

To improve system performance, optical designers make sure that thetotal aberration contribution from all surfaces taken together sums tonearly zero. Normally, such a process requires computerized analysis andoptimization. However, there are some simple guidelines that can beused to achieve this with lenses available in this catalog. This approachcan yield systems that operate at a much lower f-number than can usuallybe achieved with simple lenses.

Specifically, we will examine how to null the spherical aberration from twoor more lenses in collimated, monochromatic light. This technique will thusbe most useful for laser beam focusing and expanding.

Figure 1.31 shows the third-order longitudinal spherical aberrationcoefficients for six of the most common positive and negative lens shapeswhen used with parallel, monochromatic incident light. The plano-convexand plano-concave lenses both show minimum spherical aberration whenoriented with their curved surface facing the incident parallel beam. Allother configurations exhibit larger amounts of spherical aberration. Withthese lens types, it is now possible to show how various systems can becorrected for spherical aberration.

A two-element laser beam expander is a good starting example. In this case,two lenses are separated by a distance that is the sum of their focal lengths,so that the overall system focal length is infinite. This system will notfocus incoming collimated light, but it will change the beam diameter. Bydefinition, each of the lenses is operating at the same f-number.

The equation for longitudinal spherical aberration shows that, for two lenseswith the same f-number, aberration varies directly with the focal lengths ofthe lenses. The sign of the aberration is the same as focal length. Thus, it shouldbe possible to correct the spherical aberration of this Galilean-type beamexpander, which consists of a positive focal length objective and a negativediverging lens.

longitudinal spherical aberration (3rd order) = kf

f/#2

symmetric-concave

symmetric-convexplano-convex (reversed) plano-convex (normal)

plano-concave (reversed) plano-concave (normal)

aberrationcoefficient(k)

1.069 0.403 0.272

positive lenses

negative lenses

Figure 1.31 Third-order longitudinal spherical aberration of typical lens shapes

LSA

f/#

f/#= +

0 272 1 06912

22

. ..

f f

f

f1

2

1 0690 272

3 93= − = −..

. .




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The relatively low f numbers of these objectives is a great advantage inminimizing the length of these beam expanders. They would be particu-larly useful with Nd:YAG and argon-ion lasers, which tend to have largeoutput beam diameters.

These same principles can be utilized to create high numerical apertureobjectives that might be used as laser focusing lenses. Figure 1.32(c)shows an objective consisting of an initial negative element, followed bytwo identical plano-convex positive elements. Again, all of the elementsoperate at the same f-number, so that their aberration contributionsare proportional to their focal lengths. To obtain zero total sphericalaberration from this configuration, we must satisfy

Therefore, a corrected system should result if the focal length of thenegative element is just about half that of each of the positive lenses.In this case, f1 = 425 mm and f2 = 50 mm yield a total system focal lengthof about 25 mm and an f-number of approximately f/2. This objective,corrected to 1/6 wave, has the additional advantage of a very longworking distance.


Fundamental Optics


a) CORRECTED 4!BEAM EXPANDER

c) SPHERICALLY CORRECTED 25-mm EFL f/2.0 OBJECTIVE

f = 80 mm22.4-mm diameterplano-convex

f = 420 mm10-mm diameterplano-concave


f = 50 mm (2)27-mm diameterplano-convex

b) CORRECTED 2.7! BEAM EXPANDER


f = 54 mm32-mm diametersymmetric-convex

Figure 1.32 Combining catalog lenses for aberrationbalancing

1 069 0 272 0 272 0

0 51

1 2 2

1

2

. . .

. .

or

f f f

f

f

+ + =

= −




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FOCAL POINT (F OR F″″)

Rays that pass through or originate at either focal point must be, on theopposite side of the lens, parallel to the optical axis. This fact is the basis forlocating both focal points.

PRIMARY PRINCIPAL SURFACE

Let us imagine that rays originating at the front focal point F (and thereforeparallel to the optical axis after emergence from the opposite side of the lens)are singly refracted at some imaginary surface, instead of twice refracted (onceat each lens surface) as actually happens. There is a unique imaginarysurface, called the principal surface, at which this can happen.

To locate this unique surface, consider a single ray traced from the air onone side of the lens, through the lens and into the air on the other side.The ray is broken into three segments by the lens. Two of these are external(in the air), and the third is internal (in the glass). The external segmentscan be extended to a common point of intersection (certainly near, andusually within, the lens). The principal surface is the locus of all such

Definition of Termswww.cvimellesgriot .com


Fundamental Optics

FOCAL LENGTH ( f )

Two distinct terms describe the focal lengths associated with every lens orlens system. The effective focal length (EFL) or equivalent focal length(denoted f in figure 1.33) determines magnification and hence the imagesize. The term f appears frequently in lens formulas and in the tables ofstandard lenses. Unfortunately, because f is measured with reference toprincipal points which are usually inside the lens, the meaning of f is notimmediately apparent when a lens is visually inspected.

The second type of focal length relates the focal plane positions directlyto landmarks on the lens surfaces (namely the vertices) which are imme-diately recognizable. It is not simply related to image size but is especiallyconvenient for use when one is concerned about correct lens positioning ormechanical clearances. Examples of this second type of focal length are thefront focal length (FFL, denoted ff in figure 1.33) and the back focal length(BFL, denoted fb).

The convention in all of the figures (with the exception of a singledeliberately reversed ray) is that light travels from left to right.

f

H″

A

tc

te

F″F

B

f

fb

H

ff

ray from object at infinityray from object at infinity

front focal point

primary vertex A1

back focal point

secondary principal surfacesecondary principal point

primary principal surface

primary principal point

reversed ray locates front focal point or primary principal surface

A2 secondary vertex

optical axis

r1

r2

f = effective focal length; may be positive (as shown) or negative

ff = front focal length

fb = back focal length

A = front focus to front edge distance

B = rear edge to rear focus distance

te = edge thickness

tc = center thickness

r1 = radius of curvature of firstsurface (positive if center ofcurvature is to right)

r2 = radius of curvature of secondsurface (negative if center ofcurvature is to left)

Figure 1.33 Focal length and focal points




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points of intersection of extended external ray segments. The principalsurface of a perfectly corrected optical system is a sphere centered on thefocal point.

Near the optical axis, the principal surface is nearly flat, and for this reason,it is sometimes referred to as the principal plane.

SECONDARY PRINCIPAL SURFACE

This term is defined analogously to the primary principal surface, but it is usedfor a collimated beam incident from the left and focused to the back focalpoint F″ on the right. Rays in that part of the beam nearest the axis can bethought of as once refracted at the secondary principal surface, instead ofbeing refracted by both lens surfaces.

PRIMARY PRINCIPAL POINT (H) OR FIRST NODAL POINT

This point is the intersection of the primary principal surface with theoptical axis.

SECONDARY PRINCIPAL POINT (H″″)OR SECONDARY NODAL POINT

This point is the intersection of the secondary principal surface with theoptical axis.

CONJUGATE DISTANCES (s AND s″″)

The conjugate distances are the object distance, s, and image distance, s″.Specifically, s is the distance from the object to H, and s″ is the distancefrom H″ to the image location. The term infinite conjugate ratio refers to thesituation in which a lens is either focusing incoming collimated light or is beingused to collimate a source (therefore, either s or s″ is infinity).

PRIMARY VERTEX (A1)

The primary vertex is the intersection of the primary lens surface with theoptical axis.

SECONDARY VERTEX (A2)

The secondary vertex is the intersection of the secondary lens surface withthe optical axis.

EFFECTIVE FOCAL LENGTH (EFL, f )

Assuming that the lens is surrounded by air or vacuum (refractive index1.0), this is both the distance from the front focal point (F) to the primaryprincipal point (H) and the distance from the secondary principal point (H″)to the rear focal point (F″). Later we use f to designate the paraxial EFL forthe design wavelength (l0).

FRONT FOCAL LENGTH ( f f )

This length is the distance from the front focal point (F) to the primaryvertex (A1).

BACK FOCAL LENGTH ( fb)

This length is the distance from the secondary vertex (A2) to the rear focalpoint (F″).

EDGE-TO-FOCUS DISTANCES (A AND B)

A is the distance from the front focal point to the primary vertex of the lens.B is the distance from the secondary vertex of the lens to the rear focalpoint. Both distances are presumed always to be positive.

REAL IMAGE

A real image is one in which the light rays actually converge; if a screenwere placed at the point of focus, an image would be formed on it.

VIRTUAL IMAGE

A virtual image does not represent an actual convergence of light rays.A virtual image can be viewed only by looking back through the opticalsystem, such as in the case of a magnifying glass.

F-NUMBER (f/#)

The f-number (also known as the focal ratio, relative aperture, or speed) ofa lens system is defined to be the effective focal length divided by systemclear aperture. Ray f-number is the conjugate distance for that ray dividedby the height at which it intercepts the principal surface.

NUMERICAL APERTURE (NA)

The NA of a lens system is defined to be the sine of the angle, v1, that themarginal ray (the ray that exits the lens system at its outer edge) makeswith the optical axis multiplied by the index of refraction (n) of the medium.The NA can be defined for any ray as the sine of the angle made by that raywith the optical axis multiplied by the index of refraction:

NA = nsinv.


Fundamental Optics


f /# =f

CA. (see eq. 1.7)

(1.30)




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MAGNIFICATION POWER

Often, positive lenses intended for use as simple magnifiers are rated witha single magnification, such as 4#. To create a virtual image for viewingwith the human eye, in principle, any positive lens can be used at an infinitenumber of possible magnifications. However, there is usually a narrow rangeof magnifications that will be comfortable for the viewer. Typically, whenthe viewer adjusts the object distance so that the image appears to beessentially at infinity (which is a comfortable viewing distance for mostindividuals), magnification is given by the relationship

Thus, a 25.4-mm focal length positive lens would be a 10# magnifier.

DIOPTERS

The term diopter is used to define the reciprocal of the focal length, whichis commonly used for ophthalmic lenses. The inverse focal length of a lensexpressed in diopters is

Thus, the smaller the focal length is, the larger the power in diopters will be.

DEPTH OF FIELD AND DEPTH OF FOCUS

In an imaging system, depth of field refers to the distance in object spaceover which the system delivers an acceptably sharp image. The criteria forwhat is acceptably sharp is arbitrarily chosen by the user; depth of fieldincreases with increasing f-number.

For an imaging system, depth of focus is the range in image space overwhich the system delivers an acceptably sharp image. In other words,this is the amount that the image surface (such as a screen or piece ofphotographic film) could be moved while maintaining acceptable focus.Again, criteria for acceptability are defined arbitrarily.

In nonimaging applications, such as laser focusing, depth of focus refers tothe range in image space over which the focused spot diameter remainsbelow an arbitrary limit.



Fundamental Optics

magnification mm

in mm .=254

ff( ) (1.31)

diopters in mm=1000

ff( ). (1.32)

Technical Reference

For further reading about the definitions and formulas presentedhere, refer to the following publications:

If you need help with the use of definitions and formulaspresented in this guide, our applications engineers will bepleased to assist you.

Rudolph Kingslake, Lens Design Fundamentals(Academic Press)

Rudolph Kingslake, System Design(Academic Press)

Warren Smith, Modern Optical Engineering(McGraw Hill).

Donald C. O’Shea, Elements of Modern Optical Design(John Wiley & Sons)

Eugene Hecht, Optics (Addison Wesley)

Max Born, Emil Wolf, Principles of Optics(Cambridge University Press)

APPLICATION NOTE




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Symmetric Lens Radii (rr2 == 44rr1)

With center thickness constrained,

where, in the first form, the = sign is chosen for the square root if f ispositive, but the 4 sign must be used if f is negative. In the second form,the = sign must be used regardless of the sign of f.

Plano Lens Radius

Since r2 is infinite,

Principal-Point Locations (signed distances from vertices)

where the above sign convention applies.

For symmetric lenses (r2 = 4r1),

Paraxial Lens Formulaswww.cvimellesgriot .com

Fundamental Optics


PARAXIAL FORMULAS FOR LENSES IN AIR

The following formulas are based on the behavior of paraxial rays, which arealways very close and nearly parallel to the optical axis. In this region, lenssurfaces are always very nearly normal to the optical axis, and hence allangles of incidence and refraction are small. As a result, the sines of theangles of incidence and refraction are small (as used in Snell’s law) and canbe approximated by the angles themselves (measured in radians).

The paraxial formulas do not include effects of spherical aberrationexperienced by a marginal ray — a ray passing through the lens near itsedge or margin. All EFL values (f) tabulated in this catalog are paraxialvalues which correspond to the paraxial formulas. The following paraxialformulas are valid for both thick and thin lenses unless otherwise noted.The refractive index of the lens glass, n, is the ratio of the speed of lightin vacuum to the speed of light in the lens glass. All other variables aredefined in figure 1.33.

Focal Length

where n is the refractive index, tc is the center thickness, and the signconvention previously given for the radii r1 and r2 applies. For thin lenses,tc ≅ 0, and for plano lenses either r1 or r2 is infinite. In either case thesecond term of the above equation vanishes, and we are left with thefamiliar lens maker’s formula:

Surface Sagitta and Radius of Curvature(refer to figure 1.34)

An often useful approximation is to neglect s/2.

1 1 1 1 1

1 2

2

1 2fn

r r

n

n

t

r r= − − +

−( )

( ) c (1.33)

1 1 1 1

1 2fn

r r= − −( ) . (1.34)

r>0

(r4s )

s> 0

d

2

Figure 1.34 Surface sagitta and radius of curvature

r n f1 1= −( ) . (1.39)

( )

r n f ff t

n

n ft

n f

c

c

121

1 1 1

= − ± −⎧

⎩⎪⎪

⎫

⎭⎪⎪

= − + −⎧

⎩⎪

( )

⎪

⎫

⎭⎪⎪

A H

A H

2

1

″ =−

− + −

=−

− + −

r t

n r r t n

r t

n r r t n

c

c

c

c

2

2 1

1

2 1

1

1

( ) ( )

( ) ( )

(1.40)

(1.41)

A H A H

1 2= − ″

=− −r t

nr t nc

c

1

12 1 ( ).

(1.42)

(1.38)

r r sd

s r rd

rs d

s

2 22

22

2

2

20

2 8

= − +

= − − >

= +

( )

.

(1.35)

(1.36)

(1.37)




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If either r1 or r2 is infinite, l’Hôpital’s rule from calculus must be used.Thus, referring to Aberration Balancing, for plano-convex lenses in thecorrect orientation,

For flat plates, by letting r1 → ∞ in a symmetric lens, we obtain A1H =A2H″ = tc/2n. These results are useful in connection with the followingparaxial lens combination formulas.

Hiatus or Interstitium (principal-point separation)

which, in the thin-lens approximation (exact for plano lenses), becomes

Solid Angle

The solid angle subtended by a lens, for an observer situated at an on-axisimage point, is

where this result is in steradians, and where

is the apparent angular radius of the lens clear aperture. For an observerat an on-axis object point, use s instead of s″. To convert from steradiansto the more intuitive sphere units, simply divide Q by 4p. If the Abbé sinecondition is known to apply, v may be calculated using the arc sine functioninstead of the arctangent.

Back Focal Length

where the sign convention presented above applies to A2H″ and to the radii.If r2 is infinite, l’Hôpital’s rule from calculus must be used, whereby

Front Focal Length

where the sign convention presented above applies to A1H and to theradii. If r1 is infinite, l’Hôpital’s rule from calculus must be used, whereby

Edge-to-Focus Distances

For positive lenses,

where s1 and s2 are the sagittas of the first and second surfaces. Bevelis neglected.

Magnification or Conjugate Ratio

PARAXIAL FORMULAS FOR LENSES IN ARBITRARY MEDIA

These formulas allow for the possibility of distinct and completely arbitraryrefractive indices for the object space medium (refractive index n′), lens(refractive index n″), and image space medium (refractive index n). In thissituation, the EFL assumes two distinct values, namely f in object spaceand f″ in image space. It is also necessary to distinguish the principal pointsfrom the nodal points. The lens serves both as a lens and as a windowseparating the object space and image space media.

The situation of a lens immersed in a homogenous fluid (figure 1.35) isincluded as a special case (n = n″). This case is of considerable practicalimportance. The two values f and f″ are again equal, so that the lenscombination formulas are applicable to systems immersed in a commonfluid. The general case (two different fluids) is more difficult, and it mustbe approached by ray tracing on a surface-by-surface basis.



Fundamental Optics

f f

fr t

n r r t n

b 2

c

c

A H

= ″ + ″

= ″ −− + −

2

2 1 1( ) ( )

(1.48)

f ft

nb

c .= ″ − (1.49)

f f

fr t

n r r t nc

f 1

c

A H

= −

= +− + −

1

2 1 1( ) ( )

(1.50)

ms

s

f

s f

s f

f

= ″

=−

= ″ −

.

(1.54)

A

and

B =

f

b 2

= +

+

f s

f s

1 (1.52)

(1.53)

f ft

nf

c .= − (1.51)

HH

″ = − −−⎧

⎩⎪

⎫

⎭⎪

⎧⎨⎪⎩

⎫⎬⎪

⎭⎪t

f

n f

n

n

t

r rc

c1 1 1 2

1 2

( ) (1.44)

HH″ = −tn

c 1 1. (1.45)

vCA

= arctan2s

(1.47)

Q p v

pv

= −

=

2 1

42

2

( cos )

sin (1.46)

A H

and

A H

1

2

=

″ = −

0

t

nc .

(1.43)




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Lens Constant (kk)

This number appears frequently in the following formulas. It is anexplicit function of the complete lens prescription (both radii, tc and n′)and both media indices (n and n″). This dependence is implicit anywherethat k appears.

Effective Focal Lengths

Lens Formula (Gaussian form)

Lens Formula (Newtonian form)

Principal-Point Locations

Object-to-First-Principal-Point Distance

Second Principal-Point-to-Image Distance

Magnification

Lens Maker’s Formula

Nodal-Point Locations

Separation of Nodal Pointfrom Corresponding Principal Point

HN = H″N″ = (n″4n)/k, positive for N to right of Hand N″ to right of H″.


Fundamental Optics


f

F

fb

index n = 1.51872 (BK7)

H

A1 A2

N

ff

index n″= 1.333 (water)index n = 1 (air or vacuum)

H″ F ″N″

f ″

Figure 1.35 Symmetric lens with disparate object and image space indexes

kn n

r

n n

r

t n n n n

n r rc= ′− + ″ − ′ −

′− ″ − ′′1 2 1 2

( )( ). (1.55)

fn

kf

n

k= ″ = ″

. (1.56)

n

s

n

sk+ ″

″= . (1.57)

xx f fnn

k″ = ″ = ″

2. (1.58)

A H

A H

1c

2c

= ″− ′′

″ =− ″ ′ −

′

nt

k

n n

n r

n t

k

n n

n r

2

1.

(1.59)

(1.60)

sns

ks n= ″

″ − ″. (1.61)

″ = ″−

sn s

ks n. (1.62)

mns

n s= ″

″. (1.63)

n

f

n

fk= ″

″= . (1.64)

A N A H HN

A N A H H N

1 1

2 2

= +

″ = ″ + ″ ″.

(1.65)

(1.66)




Fundamental O

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Material Properties

Optical Coatings

For Quick Approximations

Much time and effort can be saved by ignoring the differencesamong f, fb, and ff in these formulas by assuming that f = fb = ff.Then s becomes the lens-to-object distance; s″ becomes thelens-to-image distance; and the sum of conjugate distancess=s″ becomes the object-to-image distance. This is known asthe thin-lens approximation.

APPLICATION NOTE

Physical Significance of the Nodal Points

A ray directed at the primary nodal point N of a lens appears toemerge from the secondary nodal point N″ without change ofdirection. Conversely, a ray directed at N″ appears to emergefrom N without change of direction. At the infinite conjugateratio, if a lens is rotated about a rotational axis orthogonal tothe optical axis at the secondary nodal point (i.e., if N″ is thecenter of rotation), the image remains stationary during therotation. This fact is the basis for the nodal slide method formeasuring nodal-point location and the EFL of a lens. The nodalpoints coincide with their corresponding principal points whenthe image space and object space refractive indices are equal(n = n″). This makes the nodal slide method the most precisemethod of principal-point location.

APPLICATION NOTE

Back Focal Length

Front Focal Length

Focal Ratios

The focal ratios are f/CA and f″/CA, where CA is the diameter of the clearaperture of the lens.

Numerical Apertures

Solid Angles (in steradians)

To convert from steradians to spheres, simply divide by 4p.



Fundamental Optics

n

s

n

where

and

where

sin

arcsin

sin

arcsin

v

vCA

v

vCA

=

″ ″

″ =

2

22 ″s.

Q p v

pv

vCA

Q

= −

=

=

=

where arctan2s

2 1

42

2

2

( cos )

sin

pp v

pv

vCA

(1 cos )

sin2

where 2s

2

4 ″

=

″ =″

4

arctan ..

(see eq. 1.46)

f fb 2A H .= ″ + ″ (see eq. 1.48)

f ff A H.= − 1 (see eq. 1.50)




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points will fall outside the lens boundaries. For symmetric lenses, theprincipal points divide that part of the optical axis between the verticesinto three approximately equal segments. For plano lenses, one principalpoint is at the curved vertex, and the other is approximately one-third ofthe way to the plane vertex.

Principal-Point Locationswww.cvimellesgriot .com

Fundamental Optics


Figure 1.36 indicates approximately where the principal points fall inrelation to the lens surfaces for various standard lens shapes. The exactpositions depend on the index of refraction of the lens material, and onthe lens radii, and can be found by formula. in extreme meniscus lensshapes (short radii or steep curves), it is possible that both principal

H″ F″

H″

H″F″

F″

H″F″

H″F″

H″F″

H″

F″

F″

H″

H″F″

H″F″

Figure 1.36 Principal points of common lenses




1ch FundamentalOptics Final a.qxd 6/15/2009 2:28 …... f is =for convex (diverging) ... A simple...

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Transcript of 1ch FundamentalOptics Final a.qxd 6/15/2009 2:28 …... f is =for convex (diverging) ... A simple...